the cryosphere using in-situ temperature measurements to

Using in-situ temperature measurements to estimate

saturated soil thermal properties by solving a sequence

of optimization problems

D. J. Nicolsky, V. E. Romanovsky, G. S. Tipenko

To cite this version:

D. J. Nicolsky, V. E. Romanovsky, G. S. Tipenko. Using in-situ temperature measurementsto estimate saturated soil thermal properties by solving a sequence of optimization problems.The Cryosphere, Copernicus 2007, 1 (1), pp.41-58.

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The Cryosphere

Using in-situ temperature measurements to estimate saturated soilthermal properties by solving a sequence of optimization problems

D. J. Nicolsky1, V. E. Romanovsky1, and G. S. Tipenko2

1Geophysical Institute, University of Alaska Fairbanks, PO Box 757320, Fairbanks, AK 99775, USA2Institute of Environmental Geoscience Russian Academy of Sciences, 13-2 Ulansky pereulok, PO Box 145, Moscow, Russia

Received: 1 August 2007 – Published in The Cryosphere Discuss.: 9 August 2007Revised: 17 October 2007 – Accepted: 6 November 2007 – Published: 22 November 2007

Abstract. We describe an approach to find an initial approx-imation to the thermal properties of soil horizons. This tech-nique approximates thermal conductivity, porosity, unfrozenwater content curves in horizons where no direct temperaturemeasurements are available. To determine physical prop-erties of ground material, optimization-based inverse tech-niques are employed to fit the simulated temperatures to themeasured ones. Two major ingredients of these techniquesare an algorithm to compute the soil temperature dynamicsand a procedure to find an initial approximation to the groundproperties. In this article we show how to determine the ini-tial approximation to the physical properties and present anew finite element discretization of the heat equation withphase change to calculate the temperature dynamics in soil.We successfully apply the proposed algorithm to recover thesoil properties for the Happy Valley site in Alaska using one-year temperature dynamics. The determined initial approxi-mation is utilized to simulate the temperature dynamics overseveral consecutive years; the difference between simulatedand measured temperatures lies within uncertainties of mea-surements.

1 Introduction

Recently, the Arctic Climate Impact Assessment report(ACIA, 2004) concluded that climate change is likely to sig-nificantly transform present natural environments, particu-larly across extensive areas in the Arctic and sub-Arctic.Among the highlighted potential transformations is soilwarming which can potentially cause an increase in the ac-tive layer thickness and degradation of permafrost as well ashave broader impacts on soil hydrology, northern ecosystemsand infrastructure. Since permafrost is widely distributed and

Correspondence to: D. J. Nicolsky([email protected])

covers approximately 25% of the land surface in the NorthernHemisphere (Brown et al., 1997), it is very important to un-derstand the causes affecting soil temperature regime. Oneapproach to studying soil temperature dynamics and theirdependence on climate variability is to employ mathemat-ical modeling (Goodrich, 1982; Nelson and Outcalt, 1987;Kane et al., 1991; Zhuang et al., 2001; Ling and Zhang, 2003;Oleson et al., 2004; Sazonova et al., 2004; Molders and Ro-manovsky, 2006)

A mathematical model of soil freezing/thawing is based onfinding a solution of a non-linear heat equation over a speci-fied domain, (see Andersland and Anderson, 1978; Yershov,1998, and many references therein). The domain representsground material and is divided into several horizons (e.g. anorganic matt, an organically enriched mineral soil layer, anda mineral soil layer) each with its distinct properties charac-terized by mineral-chemical composition, texture, porosity,heat capacity and thermal conductivity. By parameterizingthe coefficients in the heat equation within each horizon, itis possible to take into account temperature-dependent latentheat effects occurring when ground freezes and thaws. Thisapproach yields a realistic model of temperature dynamicsin soils. However, in order to produce quantitatively reason-able results, it is necessary to prescribe physical properties ofeach horizon.

Conventional Time Domain Reflectometry (Topp et al.,1980) and drying methods are commonly used to estimatesoil water content at shallow depths. The Time DomainReflectometry method is based on measurements of the ap-parent dielectric constant around a wave guide inserted intothe soil. It has been demonstrated that there is a relation-ship between the apparent dielectric constant and liquid wa-ter content (Topp et al., 1980) enabling robust estimationsof water content in shallow soils with homogeneous com-position. There are some difficulties however in measuringunfrozen water content of coarsely textured, heterogeneousor organically enriched soils in Arctic tundra (Boike and

Published by Copernicus Publications on behalf of the European Geosciences Union.

42 D. J. Nicolsky et al.: Estimation of thermal properties of saturated soils

Roth, 1997; Yoshikawa et al., 2004). More accurate mea-surements of the total water content (ice and water together)can be acquired by thermalization of neutrons and gammaray attenuation. This is not always suitable for Arctic re-gions as it requires transportation of radioactive equipment toremote locations (Boike and Roth, 1997). An alternative tothe above-mentioned methods and also to a number of others(Schmugge et al., 1980; Tice et al., 1982; Ulaby et al., 1982;Stafford, 1988; Smith and Tice, 1988) is the use of inversemodeling techniques. These techniques estimate the watercontent and other thermal properties of soil using in-situ tem-perature measurements and by exploiting the mathematicalmodel.

A variety of inverse modeling techniques that recover thethermal properties of soil are known. Many of them relyon the commonly called source methods (Jaeger and Sass,1964), in which temperature response due to heating is mea-sured at a certain distance from the heat source. The temper-ature response and geometry of the probe are used to com-pute the thermal properties by either direct or indirect meth-ods. In the direct methods, the temperature measurementsare explicitly used to evaluate the thermal properties. In theindirect methods, one minimizes a discrepancy between themeasured and the synthetic temperatures, the latter computedmathematically by exploiting the heat equation in which thecoefficients are parameterized according to the specified ther-mal properties.

Application of direct methods such as the Simple FourierMethods (Carson, 1963), Perturbed Fourier Method (Hur-ley and Wiltshire, 1993), and the Graphical Finite DifferenceMethod (McGaw et al., 1978; Zhang and Osterkamp, 1995;Hinkel, 1997) yield accurate results for the thermal diffusiv-ity (the ratio of the thermal conductivity and the heat capac-ity), only when water does not undergo the phase change.Despite the fact that the direct methods are well establishedfor the heat equation without the phase change, no univer-sal framework exists in the case of the soil freezing/thawingbecause the heat capacity and thermal conductivity dependstrongly on the temperature in this case.

A common implementation of the indirect methods uses ananalytical or numerical solution of the heat equation to eval-uate the synthetic temperature. Due to strong non-linearities,the analytical solution of the heat equation is known onlyfor a limited number of cases (Gupta, 2003), whereas nu-merical solutions are typically computable. Given a nu-merical solution computed by finite difference (Samarskiiand Vabishchevich, 1996) or finite element (Zienkiewicz andTaylor, 1991) methods, one can minimize a cost function,J ,which measures a discrepancy between the measuredTm andsyntheticTc temperatures. The typical expression for the costfunction,J , is given by

J (C) ≈

∫ te

ts

(Tm(xi, t)− Tc(xi, t; C))2dt. (1)

Here, the quantityC is the control vector that is a set of pa-rameters defining soil properties of each soil horizon. Thesynthetic temperature,Tc, is computed by the mathematicalmodel parameterized by variables inC at some depthsxi overthe time interval[ts, te].

In this article, we deal with optimization techniques thatfind soil properties by minimizing the cost function (1).Commonly, the cost functionJ is minimized iteratively start-ing from an initial approximationC0 to the parametersC(Thacker and Long, 1988). Since the heat equation is non-linear, in general there are several local minima. Hence, itis important that the initial approximation lies in the basin ofattraction of a proper minimum (Avriel, 2003).

We present a semi-heuristic algorithm to determine an ini-tial approximationC0, for use as the starting point in mul-tivariate minimization of cost functions such as (1). In thisarticle, we use in-situ measured temperatureTm to formulatethe cost functionJ . We construct the initial approximationby minimizing cost functions over specifically selected timeintervals[ts, te] in a certain order. For example, first, we pro-pose to find thermal conductivity of the frozen soil using thetemperature collected during winter, and then use these val-ues to find properties of the thawed soil. In order to minimizethe cost function it is necessary to compute the temperaturedynamics multiple times for various control vectorsC. Sincean analytical solution of the non-linear heat equation is notgenerally available, we use a finite element method to find itssolution. To compute latent heat effects, we propose a newfixed grid technique to evaluate the latent heat terms in themass (compliance) matrix using enthalpy formulation. Ourtechniques do not rely on temporal or spatial averaging ofenthalpy, but rather evaluate integrals directly by employinga certain change of variables. An advantage of this approachis that it reduces the numerical oscillation of the temperaturedynamics at locations near 0◦C isotherm.

The structure of this article is as follows. In Sect. 2, wedescribe a commonly used mathematical model of tempera-ture changes in the active layer and near surface permafrost.In Sect. 3, we outline a finite element discretization of theheat equation with phase change. In Sect. 4, we introducemain definitions, notations and state the variational approachto find the thermal properties. In Sect. 5, we provide an algo-rithm to construct an initial approximation to thermal proper-ties. In Sect. 6, we apply our method to estimate the thermalproperties and the coefficients determining the unfrozen wa-ter content at a site located in Alaska. In Sect. 7, we statelimitations and shortcomings of the proposed algorithm. Fi-nally, in Sect. 8, we provide conclusions and describe mainresults.

2 Modeling of soil freezing and thawing

For many practical applications, heat conduction is a dom-inant process, and hence the soil temperatureT , [◦C] can

The Cryosphere, 1, 41–58, 2007 www.the-cryosphere.net/1/41/2007/

D. J. Nicolsky et al.: Estimation of thermal properties of saturated soils 43

be simulated by a 1-D heat equation with phase change(Carslaw and Jaeger, 1959):

C∂

∂tT (x, t)+ L

∂

∂tθl(T , x) =

∂

∂xλ∂

∂xT (x, t), (2)

where x∈[0, l], t∈[0, τ ]; the quantities C=C(T , x)

[Jm−3 K−1] andλ=λ(T , x) [Wm−1 K−1] stand for the volu-metric heat capacity and thermal conductivity of soil, respec-tively; L [Jm−3] is the volumetric latent heat of fusion ofwater, andθl is the volumetric liquid water content. We notethat this equation is applicable when migration of water isnegligible, there are no internal sources or sinks of heat, frostheave is insignificant, and there are no changes in topographyand soil properties in lateral directions. Typically, the heatEq. (2) is supplemented by Dirichlet, Neumann, or Robinboundary conditions specified at the ground surface,x=0,and at the depthl (Carslaw and Jaeger, 1959). In geothermalstudies, a Neumann boundary condition is typically set at thedepthl. In this study we use the measured temperaturesTuandTl to set the Dirichlet boundary conditions at depthsx=0andx=l, respectively, i.e.T (0, t)=Tu(t), T (l, t)=Tl(t). Inorder to calculate the temperature dynamicsT (x, t) at anytime t∈[0, τ ], Eq. (2) is supplemented by an initial condi-tion, i.e. T (x,0)=T0(x), whereT0(x) is the temperature atx∈[0, l] at timet=0.

In certain conditions such as waterlogged Arctic lowlands,soil can be considered a porous media fully saturated withwater. The fully saturated soil is a multi-component sys-tem consisting of soil particles, liquid water, and ice. It isknown that the energy of the multi-component system is min-imized when a thin film of liquid water (at temperature below0◦C) separates ice from the soil particles (Hobbs, 1974). Afilm thickness depends on soil temperature, pressure, miner-alogy, solute concentration and other factors (Hobbs, 1974).One of the commonly used measures of liquid water belowfreezing temperature is the volumetric unfrozen water con-tent (Williams, 1967; Anderson and Morgenstern, 1973; Os-terkamp and Romanovsky, 1997; Watanabe and Mizoguchi,2002). It is defined as the ratio of liquid water volume in arepresentative soil domain at temperatureT to the volume ofthis representative domain and is denoted byθl(T ). Thereare many approximations toθl in the fully saturated soil(Lunardini, 1987; Galushkin, 1997). The most common ap-proximations are associated with power or exponential func-tions. Based on our positive experience in (Romanovsky andOsterkamp, 2000), we parameterizeθl by a power functionθl(T )=a|T |−b; a, b>0 for T<T∗<0◦C (Lovell, 1957). TheconstantT∗ is called the freezing point depression (Hobbs,1974), and from the physical point of view it means that icedoes not exist in the soil ifT>T∗. In thawed soils (T>T∗),the amount of water in the saturated soil is equal to the soilporosityη, and hence the functionθl(T ) can be extended toT>T∗ asθl(T )=η. Therefore, we assume that

θl(T , x)=η(x)φ(T , x), φ=

{

1, T ≥ T∗

|T∗|b|T |−b, T < T∗

, (3)

Fig. 1. Typical volumetric content of the unfrozen liquid water insoils as a function of temperature. The curve marked by triangles isassociated with soils in which all water is bound in soil pores, andhence the water content gradually decreases with decreasing tem-perature in◦C. To compute this curve we used parametrization (3)in whichT∗=−0.03◦C andb=0.3. The curve marked by circles isrelated to soils in which some percentage of water is not bound tothe soil particle and changes its phase at the temperatureT∗, whileother part of liquid water is bound in soil pores and freezes gradu-ally as the temperature decreases.

whereφ=φ(T , x) represents the liquid pore water fraction,andT is in ◦C, see the curve marked by triangles in Fig. 1.Note that the constantsT∗ andb are the only parameters thatspecify dependence of the unfrozen liquid water content ontemperature. For example, small values ofb describe the liq-uid water content in some fine-grained soils, whereas largevalues ofb are related to coarse-grained materials in whichalmost all water freezes at the temperatureT∗. The limitingcase in which all water freezes at the temperatureT∗ is as-sociated with phase change between water and ice (no soilparticles). This limiting case is commonly called the clas-sical Stefan problem and is represented by extremely largevalues ofb in (3).

In this article, we use the following notation and defini-tions. We abbreviate by lettersi, l ands, ice, liquid water,and the soil particles, respectively. We express thermal con-ductivity λ of the soil and its apparent volumetric heat ca-pacityCapp according to (de Vries, 1963; Sass et al., 1971)as

λ(T )=λθss λθi (T )i λ

θl(T )l , Capp(T )=C(T )+L

dθl(T )

dT(4)

C(T ) = θi(T )Ci + θl(T )Cl + θsCs (5)

whereC is called the volumetric heat capacity of the soil.Here, the constantsCk, λk, k∈{i, l, s} are the volumetric heat

www.the-cryosphere.net/1/41/2007/ The Cryosphere, 1, 41–58, 2007


Table 1. A typical thickness of soil layers and commonly occurring range of thermal properties in a cryosol soil at the North Slope, Alaska.

Layer Layer thickness Thermal conductivity Porosity, Coefficient in (3)in the frozen state,λf η b

Moss or organic layer 0.05 [0.1,0.7] [0.1,0.7] [1.0,0.5]

Mineral-organic mixture 0.20 [0.9,1.6] [0.2,0.6] [0.8,0.5]

Mineral soil >1.0 [1.3,2.4] [0.2,0.4] [0.7,0.5]

capacity and thermal conductivity of thek-th constituent at0◦C, respectively. The quantityθk, k∈{i, l, s} is the volumefraction of each constituent. Exploiting the relationsθs=1−η

andθi=η−θl , we introduce notation for the effective volu-metric heat capacitiesCf andCt , and the effective thermalconductivitiesλf andλt of soil for frozen and thawed states,respectively. Therefore formulae (4) and (5) yield

Capp=C+Ldθl

dT, C=Cf (1−φ)+Ctφ, λ=λ

1−φf λ

φt , (6)

where

λf=λ1−ηs λ

ηi , λt=λ

1−ηs λ

ηl

Cf=Cs(1−η)+Ciη, Ct=Cs(1−η)+Clη.

For most soils, seasonal deformation of the soil skeleton isnegligible, and hence temporal variations in the total soilporosity η for each layer are insignificant. Therefore, thethawed and frozen thermal conductivities for the fully satu-rated soil satisfy

λt

λf=

[λl

λi

]η

. (7)

It is important to emphasize that evaporation from the groundsurface and from within the upper organic layer can causepartial saturation of upper soil horizons (Hinzman et al.,1991; Kane et al., 2001). Therefore, formula (7) need nothold within live vegetation and organic soil layers, and pos-sibly within organically enriched mineral soil (Romanovskyand Osterkamp, 1997).

In this article, we approximate the coefficientsCapp, λ ac-cording to (6), where the thermal propertiesλf , λt , Cf , Ctand parametersη, T∗, b are constants within each soil hori-zon. Table 1 lists typical soil horizon geometry, commonlyoccurring ranges for the porosityη, thermal conductivityλfand the values ofb parameterizing the unfrozen water con-tent.

3 Solution of the heat equation with phase change

3.1 A review of numerical methods

In order to solve the inverse problem one needs to compute aseries of direct problems, i.e. to obtain the temperature fields

for various combinations of thermal properties. A numberof numerical methods (Javierre et al., 2006) exist to computetemperature that satisfies the heat equation with phase change(2). These methods vary from the simplest ones which yieldinaccurate results to sophisticated ones which produce ac-curate temperature distributions. The highly sophisticatedmethods explicitly track a region where the phase changeoccurs and produce a grid refinement in its vicinity, andtherefore take significantly more computational time to ob-tain temperature dynamics. Implementing such complicatedmethods is not always necessary, since an extremely accuratesolution is not particularly important when the mathematicalmodel describing nature is significantly simplified.

In this subsection, we briefly review several fixed gridtechniques (Voller and Swaminathan, 1990) that accuratelyestimate soil temperature dynamics and easily extend tomulti-dimensional versions of the heat Eq. (2). These meth-ods provide the solution for arbitrary temperature-dependentthermal properties of the soil and do not explicitly track thearea where the phase change occurs. Recall that in soils thephase change occurs at almost all sub-zero temperatures. Acornerstone of the fixed grid techniques is a numerical ap-proximation of the apparent heat capacityCapp. A varietyof the approximation techniques can be found in (Voller andSwaminathan, 1990; Pham, 1995) and references therein. Ingeneral, two classes of them can be identified. The first classis based on temperature/coordinate averaging (Comini et al.,1974; Lemmon, 1979) of the phase change. Here, the appar-ent heat capacity is approximated by

Capp =∂H

∂x

(∂T

∂x

)−1, (8)

where

H =

∫ T

0CappdT ,

is the enthalpy. The second class of methods is based ontemperature/time averaging (Morgan et al., 1978). In thisapproach,

Capp =Hcurrent−Hprevious

Tcurrent− Tprevious, (9)

where subscripts mark time steps at which the values ofH

andT are calculated. Although these methods have been pre-sented in the context of large values ofb in (3), it is noted that



they work best in the case of a naturally occurring wide phasechange interval. Also, it is important to note that the approx-imation (8) is not accurate for near zero temperature gradi-ents. In the case when the boundary conditions are givenby natural variability (several seasonal freezing/thawing cy-cles), near zero gradients at some depths may occur for sometime intervals. Hence, the temperature dynamics calculatedby using (8) can have large computational errors.

An alternative fixed grid technique can be developed byrewriting the heat equation (2) in a new form:

∂H

∂t=

∂

∂xλ∂

∂xT , T = T (H), (10)

resulting in the enthalpy diffusion method (Mundim andFortes, 1979). Advantages of discretizing (10) is that thetemperatureT=T (H) is a smooth function of enthalpyHand hence one can compute all partial derivatives. However,for soils with a sharp boundary between thawed and com-pletely frozen states, the enthalpyH becomes a multivariatefunction when temperatureT nearsT∗. Therefore, solutionof (10) results in that the front becomes artificially stretchedover at least one or even several finite elements.

In this article, we propose a fixed grid technique that ap-plies the basic finite element method (Zienkiewicz and Tay-lor, 1991) to Eq. (2). Finite element discretization of

L∂θl

∂t= L

dθl

dT

∂T

∂t

in the left hand side of (2) results in

(

∫ x1

x0

ψi(x)ψj (x)Ldθl

dT

(

T (x, t))

dx) dTj

dt, (11)

where ψi(x) and ψi(x) are two piecewise linear basisfunctions at nodesi and j , respectively, Tj (t) is thevalue of temperature at thej -th node at timet , andT (x, t)=

∑

i ψi(x)Ti(t). We propose to evaluate this typeof integrals using the unfrozen liquid water contentθl as theintegration variable, i.e.∫ x1

x0

ψ(x)Ldθl

dT

(

T (x, t))

dx = L

∫ θ1

θ0

ψ(

T (θl, t))

dθl, (12)

whereψ=ψiψj , andθ0=θl(T (x0, t)) andθ1=θl(T (x1, t)).This substitution allows precise computation of the latentheat effect for arbitrary grid cells, since it is parameterizedby the limits of integrationθ0, θ1, instead of being calcu-lated using the rapidly varying functiondθl

dT(T ) on the el-

ement[x0, x1] by a quadrature rule. As a consequence ofthe proposed substitution, evaluation of the integral in (11)may not to yield the right result unless the functionT (θl)must be monotonically increasing for allθl<η, andT (x, t)be monotonous on[x0, x1] at time t . Figure 1 shows twoinstances of the unfrozen water content curves frequently oc-curring in nature. The curve marked by circles is associatedwith soils in which free water freezes prior to freezing of the

bound liquid water in soil pores. The free water is associ-ated with a vertical line atT=T∗ whereas the bound water isrepresented by a smooth curve atT<T∗. The curve markedby triangles reflects soil in which all water is bounded in soilpores and can be parameterized by (3) used in our modeling.

3.2 Finite element formulation

Let us consider a triangulation of the interval[0, l] by a setof nodes{xi}ni=1. With each nodexi , we associate a contin-uous functionψi(x) such thatψi(xj )=δij . We will refer to{ψi}

ni=1 as the basis functions on the interval[0, l]. Hence,

the temperatureT (x, t) on [0, l] is approximated by a linearcombination:T (x, t)=

∑ni=1 Ti(t)ψi(x), whereTi=Ti(t) is

the temperature at the nodexi at the timet . Substituting thislinear combination into (2), multiplying it byψj and thenintegrating over the interval[0, l], we obtain a system of dif-ferential equations (Zienkiewicz and Taylor, 1991):

M(T)d

dtT(t) = −K(T)T(t), (13)

whereT≡T(t)=[T1(t) T2(t) . . . Tn(t)]t is the vector of tem-

peratures at nodes{xi}ni=1 at timet . Here, then×n matricesM(T)={mij }

nij=1 andK(T)={kij }

nij=1 are mass and stiffness

matrices, respectively. Entry-wise they are defined as

mij=

∫ l

0C(T , x)ψiψjdx+L

∫ l

0

dθl

dTψiψjdx (14)

kij=

∫ l

0λ(T , x)

dψi

dx

dψj

dxdx. (15)

The fully implicit scheme is utilized to discretize (13) withrespect to time. Denoting bydtk the time increment at thek-th moment of timetk, one has[

M k + dtkK k]

Tk = M kTk−1, k > 1 (16)

where Tk=T(tk), K k=K(Tk), M k=M(Tk). We imposeboundary conditions atx=0 and some depthx=l by spec-ifying T1(tk)=Tu(tk) andTn(tk)=Tl(tk).

GivenTk−1, we find the solutionTk of (16) by Picard it-eration (Kolmogorov and Fomin, 1975). The iteration pro-cess starts from the initial guessTk0 = Tk−1 that is used tocompute temperatureTk1 at the first iteration. At iterations,we computeTks and terminate iterations atse when a cer-tain convergence condition is met. The value ofTks is usedto evaluate the matricesM k

s=M(Tks ), andK ks=K(Tks ). In

turn, these are utilized to compute thes+1 iterationTks+1 byequating

[M ks + dtkK k

s]Tks+1 − M ksTk−1 = 0. (17)

At each iteration the convergence conditionmaxk |T s+1

k (tk)−Tsk (tk)|≤ǫ is checked. If it hold, the

iterations are terminated atse=s+1. If the number ofiterations exceeds a certain predefined number, the timeincrementdtk is halved and the iterations start again. Please,note that the convergence is guaranteed if the time incrementdtk is small enough.



Fig. 2. Comparison of analytical (stars) and numerical solutions.Initially, the soil has−5◦C temperature, and at the timet=0, thetemperature at its upper boundary is changed to 1◦C. At the lowerboundary located at 5 m depth, zero flux boundary condition is spec-ified. On the left plot, we show a location of the 0◦C isotherm calcu-lated for a uniform spatial discretizations with 0.1 m grid element.The numerical solutions are computed by the proposed method (cir-cles) and by the scheme using the lumped approach with temporalenthalpy averaging (squares). In the right plot, we show an enlargedarea within the dotted rectangle and a location of the 0◦C isothermcalculated for a uniform spatial discretizations with 0.1 m (filled)and 0.01 m (hollow) grid elements.

3.3 Computation of the mass matrix

One of the obstacles to obtain a finite dimensional approxi-mation that accurately captures the temperature dynamics isrelated to evaluation of the mass matrixM . Since the basisfunctionψi does not vanish only on the interval[xi−1, xi+1],the matrixM is tri-diagonal. Therefore, to compute itsi-throw we evaluate∫ l

0

dθl

dTψj (x)ψi(x)dx j = i − 1, i, i + 1, (18)

wherej stands for the column index. For the sake of brevity,we consider the first integral (j=i−1) in (18). This restrictsus only to the grid element[xi−1, xi], yielding∫ l

0

dθl

dTψi−1(x)ψi(x)dx =

∫ xi

x−1

dθl

dTψi−1(x)ψi(x)dx. (19)

We recall that in the standard finite element method, the tem-perature on the interval[xi−1xi] is approximated by

T (x, t)=ψi−1(x)Ti−1(t)+ψi(x)Ti(t), (20)

for anyx ∈ [xi−1, xi] and fixed moment timet . Here,ψi andψi−1 are piece-wise linear functions satisfyingψi−1=1−ψion[xi−1, xi]. For allx ∈ [xi−1, xi], we can compute the tem-peratureT from (20) and values ofTi , Ti−1. Note that in the

case of1Ti=0, we can compute (19) directly sincedθl/dTis constant over[xi−1xi]. However, if1Ti=Ti−Ti−1 6=0,then we can consider an inverse function, that is,x is takenas a function ofT to obtain∫ xi

xi−1

dθl

dTψi−1ψidx =

xi − xi−1

(1Ti)3

∫ xi

xi−1

dθl

dT(Ti − T )(T − Ti−1)dT

Therefore∫ l

0

dθl

dTψi−1ψidx =

xi − xi−1

(1Ti)3

∫ θi

θi−1

(T − Ti)(Ti−1 − T )dθ,

(21)

whereθi−1=θl(T (xi−1, t)) andθi=θl(T (xi, t)). Note that in(21) the temperatureT is a function of the liquid water con-tentθl , i.e.T=θ−1

l (θl). Therefore, returning back to (18), wehave that each of the integrals in (18) is a linear combinationof the typeβ2A2+β1A1+β0A0, where

Ak =

∫ θi

θi−1

[θ−1l (z)]kdz, k = 0,1,2.

The constants{βk} are easily computable ifθl(T ) is given by(3).

3.4 Evaluation of the proposed method

To test the proposed method, we compare temperature dy-namics computed by the proposed method with an analyticalsolution of the heat Eq. (2) in whichb→∞. This analyticalsolution is called Neumann solution (Gupta, 2003) and is typ-ically used to verify numerical schemes. In the proposed nu-merical scheme the mass matrixM is tri-diagonal, and hencethis scheme is called consistent. Other commonly utilizednumerical schemes are called mass lumped (Zienkiewicz andTaylor, 1991) since they employ the diagonal mass matrix:

M = diag(Capp,1

∫ 1

0ψ1dx, . . . , Capp,n

∫ 1

0ψndx). (22)

Here,Capp,i is the value of the apparent heat capacityCappatthe i-th node computed either by spatial (8) or temporal (9)averaging of latent heat effects.

In Fig. 2, we compare temperature dynamics computed bythe proposed consistent and a typical mass lumped scheme.We plot a location of the 0◦C isotherm for several spatial dis-cretizations, i.e. the distance1xi between two neighboringnodesxi and xi−1 is 0.1 or 0.01 m. In this figure we seethat the location of the 0◦C isotherm calculated by numeri-cal schemes lies within1xi bound near the analytical solu-tion. However, temporal dynamics of the location of the 0◦Cisotherm differ among methods. In the solution (squares)computed by the mass lumped approach with temporal en-thalpy averaging (TA), dynamics of the 0◦C isotherm hassome irregularities, i.e. the freezing front either advancingtoo fast or too slow. In average, however this algorithm pro-duces good results. Our proposed consistent method (circles)



Fig. 3. Computed soil temperature dynamics at 0.3 m depth. Uni-form 0.01 and 0.1 m meshes are used to compute temperatures bythe consistent (circles) and mass lumped approaches, respectively.The spatial (SA) and temporal (TA) enthalpy averaging in lumpedschemes are marked my triangles and squares, respectively. Ini-tially the temperature is zero, the upper boundary condition is givenby Dirichlet type boundary condition with a slowly varying sinusoidhaving the amplitude of 3◦C and the period of there years; zero heatflux is specified at 2 m depth.

gives a better solution and smoother rate of advancing of the0◦C isotherm, see Fig. 2, left.

In Fig. 3, we compare temperature dynamics computed bytwo mass lumped approaches exploiting spatial (8) and tem-poral (9) enthalpy averaging. A warm bias in the tempera-ture computed by the spatial averaging of the enthalpy is dueto computational errors occurring when the temperature gra-dient is approximately zero at some depth. Our experienceshows that this difference appears regardless of decreasingthe toleranceǫ between iterations in (17). We note that in allabove numerical experiments a finite element computer codeis the same except for a part associated with computation ofmass matrix, i.e. consistent (18) or mass lumped (22). Thesenumerical experiments show that the straight-forward masslumped schemes are typically inferior to consistent ones.

Since our method (14) is based on the consistent approach(the mass matrixM is the tri-diagonal one), the numericalsolution oscillates if the time stepsdtk are too small (Pin-der and Gray, 1977). For a fixed time stepdtk, the oscilla-tions disappear if the spatial discretization becomes fine, i.e.the inequalitymij+dtkkij<0 holds wheni 6=j (Ciarlet, 2002;Dalhuijsen and Segal, 1986). It is shown that these oscilla-tions occur due to violation of the discrete maximum princi-ple (Rank et al., 1983). Therefore, to avoid the oscillationsin the numerical solution (Dalhuijsen and Segal, 1986), wepropose either to use sufficiently large time steps (for which

Fig. 4. Temperature dynamics at 1 m depth computed by the pro-posed consistent (circles) and the mass lumped schemes (stars). Themass lumped scheme is based on (23). In order to emphasis numer-ical oscillations occurring in the case of small time steps in the con-sistent approach, we use a uniform grid with 0.1 m grid elements.The oscillations are due to violation of the discrete maximum prin-ciple in the consistent scheme during active phase change processes.The initial and boundary conditions are the same as stated in captionof Fig. 3.

the formula can be found in the above cited references) or toexploit the following regularization. We construct a lumpedversionM={mij } of the mass matrixM given by

mii =∑

j

mij (23)

and substituteM for M in (16). Comparison of temperaturedynamics computed employing the proposed consistentMdefined by (16) and its mass lumped modificationM definedby (23) is shown in Fig. 4. The numerical oscillations near0◦C disappear in the temperature dynamics computed by theproposed mass lumped approach (see Fig. 4). In Fig. 5, wecompare the proposed mass lumped approach (stars), and theone based on temporal enthalpy averaging (squares) by (8).This figure shows that the numerical scheme using tempo-ral averaging of the enthalpy produces larger oscillation thanour solution. This comparison reveals that the proposed masslumped approach (23) reduces some numerical oscillationsand follows the “exact” solution (computed by the consis-tent approach with a fine spatial discretization) more closelythan the solution computed by the lumped approach exploit-ing (8).

In conclusion, we state that if a spatial discretization is fineand time steps are sufficiently large (Pinder and Gray, 1977)then the consistent schemes do not show numerical oscilla-tions, and hence they should be utilized. In the case of a



Fig. 5. Temperature dynamics at 1 m depth computed by the consis-tent approach (circles), the proposed mass lumped approach (stars)and the mass lumped approach with temporal enthalpy averaging(squares). The temperatures computed mass lumped approach arefound on uniform grid with 0.1 m grid elements, whereas in the con-sistent approach, the length of grid elements is 0.01 m. The initialand boundary conditions are the same as stated in caption of Fig. 3.

coarse spatial discretization, consistent schemes can violatethe discrete maximum principle, and hence the mass lumpedschemes are more attractive. In this article, we construct afine spatial discretization and use the proposed consistent ap-proach, while restricting the time steptk from below.

4 Variational approach to find the soil properties

In this section, we provide definitions and describe maincomponents of the indirect method used to find the soil prop-erties by minimizing the cost function outlined in (1).

We define the controlC as a set consisting of thermal con-ductivitiesλ(i)t , λ

(i)f , heat capacitiesC(i)t , C

(i)f and parameters

η(i), T(i)∗ , b(i) describing the unfrozen water content for each

soil horizoni=1, . . . , n, or

C = {C(i)f , C

(i)t , λ

(i)t , λ

(i)f , η

(i), T (i)∗ , b(i)}ni=1, (24)

wheren is the total number of horizons. We say that a solu-tion of the direct problem for the controlC is T (x, t; C) andis defined by the set

T (x, t; C) = {T (xi, t) : i = 1, . . . , m; t ∈ [0, τ ]}, (25)

where{xi}mi=1 is a set ofm fixed distinct points on[0,l]. In

(25), theT (xi, t) are point-wise values of temperature dis-tributions satisfying (2) in which thermal properties of eachhorizon are given according toC.

The counterpart ofT (x, t; C) is the dataTD(x, t) definedby a set of measured temperature at the same depths{xi}

mi=1

and the same time interval[0, τ ]. Since the dataTD(x, t) andits model counterpartT (x, t; C) are given on the same set ofdepths and time interval, we can easily compute a discrep-ancy between them, usually measured by the cost function

J (C)=1

m(ts−te)

m∑

i=1

1

σ 2i

∫ te

ts

(TD(xi, t)−T (xi, t; C))2dt. (26)

Here,ts, te ∈ [0, τ ] andσi stands for an uncertainty in mea-surements by thei-th sensor. In our measurements all tem-perature sensors assume the same precision, so all of{σi} areequal. Given a way to measure this discrepancy as in (26) wecan finally formulate an inverse problem.

For the given dataTD(x, t), we say that the controlC∗

is a solution to an inverse problem if discrepancy betweenthe data and its model counterpart evaluated atC∗ is mini-mal (Alifanov, 1995; Alifanov et al., 1996; Tikhonov et al.,1996). That is,

J (C∗) = minC

J (C).

To illustrate steps which are necessary to solve this inverseproblem and find an optimalC∗ we provide the followingexample. To formulate the inverse problem one has to havethe measured temperaturesTD(x, t). For the sake of this ex-ample, we replace the dataTD(x, t) by a synthetic tempera-tureTS(x, t) = T (x, t; C

′) (a numerical solution of the heatEq. (2) for the known combinationC′ of the thermal proper-ties):

C′=

C(1)f =1.6×106, C

(1)t =2.1×106, λ

(1)f =0.55, λ(1)t =0.14, η(1)=0.30, b(1)=0.9, T (1)∗ =−0.03

C(2)f =1.7×106, C

(2)t =2.3×106, λ

(2)f =0.90, λ(2)t =0.66, η(2)=0.30, b(2)=0.6, T (2)∗ =−0.03

C(3)f =1.8×106, C

(3)t =2.6×106, λ

(3)f =1.90, λ(3)t =1.25, η(3)=0.25, b(3)=0.8, T (3)∗ =−0.03

.

The initial and boundary conditions in all calculations arefixed and given by in-situ temperature measurements in 2001and 2002 at the Happy Valley site located in the AlaskanArctic. We compute the temperature dynamics for a soilslab with dimensions[0.02,1.06] between 21 July 2001 and6 May 2002, and evaluate the cost function at{xi}i={0.10,0.17, 0.25, 0.32, 0.40, 0.48, 0.55, 0.70, 0.86 m. Uniformlydistributed noise on[−0.04,0.04] was added toTS(x, t), tosimulate noisy temperature data recorded by sensors (preci-sion of the sensor is 0.04◦C). The boundaries between thehorizons lie at 0.10 and 0.20 m depth.

We find a controlC′ that minimizes the cost functionJdefined by (26) in whichTD(x, t)=TS(x, t). For the sakeof simplicity, we assume that all variables inC′ are knownexcept for the pairλ(2)f ,η

(3). Therefore, the problem of find-ing this pair can be solved by minimizing the cost functionJ on (λ(2)f ,η

(3)) plane as follows. We compute temperature

dynamics for various combinations ofλ(2)f ,η(3) and plot iso-

lines ofJ , see Fig. 6. The point on(λ(2)f ,η(3)) plane where

the cost function is minimal gives the sought values ofλ(2)f

andη(3). The location of the minimum coincides with val-uesλ(2)f =0.9, η(3)=0.25, which were used to generate thesynthetic data.



In the above example, the control had only two unknownvariablesλ(2)f , η

(3) and we minimized the corresponding costfunction. Usually, a majority of variables in the controlC isunknown, and hence multivariate minimization is required.Since computation of the cost function for all possible real-izations of the control on the discrete grid is extremely time-consuming, various iterative techniques are used (Fletcher,2000).

We note that if the cost function has several minima dueto non-linearities of the heat Eq. (2) and if the initial ap-proximationC0 is arbitrary then the iterative algorithm canconverge to an improper minimum. Nevertheless, with theinitial approximationC0 within the basin of attraction of theglobal minimum, the iterative optimization method shouldconverge to the proper minimum even if the model is nonlin-ear (Thacker, 1989). Consequently, proper determination ofan initial approximationC0 is important.

After selection of the initial approximationC0, the nextstep is to minimize the cost functionJ (C) with respect to allparameters inC. There is a great variety of iterative methodsthat minimizeJ (C). The majority of them rely on compu-tation of the gradient∇J (C) of the cost function. The com-putation of∇J (C) is a complicated problem and is out ofthe scope of this article. An interested reader is referred to(Alifanov et al., 1996; Permyakov, 2004) and to referencestherein. Since in this article we are primarily concerned withevaluation of the initial approximation to the thermal prop-erties, we use the following universal algorithm to minimizethe cost function.

We look for the minimum of the cost function by thesimplex search method described in (Lagarias et al., 1998),which is a direct search method (Bazaraa et al., 1993). Ina two and three dimensional spaces, the simplex is a trian-gle or a pyramid, respectively. At each iteration the value ofthe function computed at the point, being in or near the cur-rent simplex, is compared with the function’s values at thevertices of the simplex and, usually, one of the vertices is re-placed by the new point, giving a new simplex. The iterationprocesses is continued until the simplex sizes are less than ana priori specified tolerance. At the final iteration, we obtainthe setC of parameters that determine the thermal properties,porosity and coefficients specifying the unfrozen water con-tent for each soil horizon. However, we note that this algo-rithm typically converges to the minimum slower than otheralgorithms that require calculations of the gradient (Dennisand Schnabel, 1987).

5 Selection of an initial approximation

Selection of a proper initial approximationC0 is an impor-tant problem, since the proper choice ofC0 ensures that theminimization procedure converges to a global minimum. Inthis section we describe how to select a proper initial approx-imation by considering several simpler subproblems.

Fig. 6. Isolines of the cost functionJ (C) computed using the syn-thetic temperature dataTS. The minimum of the cost function is

marked by the start and is located atλ(2)f

=0.9 andη(3)=0.25, which

is coincide with the values ofλ(2)f

, η(3) used to computeTS.

5.1 General methodology

We begin by noting that in the natural environment, the ther-mal properties and the water content are confined withina certain range depending on soil texture and mineralogy.Therefore, the coefficients in (2) and hence their initial ap-proximations lie within certain limits. To ensure better deter-mination of the initial approximationC0, we employ an algo-rithm similar to coordinate-wise searching method (Bazaraaet al., 1993). In this method, one looks for a minimum alongone coordinate, keeping other coordinates fixed, and thenlooks for the minimum along another coordinate keeping oth-ers fixed and so on.

We propose to look for a minimum with respect to somesubset of parameters inC, followed by a search along otherparameters inC and so on. In details, our approach is formu-lated in five steps:

1. Select several time intervals{1k} in the period of ob-servations[0, τ ]

2. Associate a certain subsetCj of parametersC with each1j . The subsetCj is such that the temperature dynam-ics over the period1j is primarily determined byCjand depend much less on changes in any other parame-ters inC.

3. Select a certain pair{1j ,Cj }, and look for a locationof the minimum of the cost functionJ (C) keeping allparameters inC except forCj fixed.

4. Update values ofCj in the controlC by the results ob-tained at Step 3.

5. Select another pair{1i,Ci} that is different from thepair {1j ,Cj } at the previous step. Goto Step 3 and re-peat for the pair{1i,Ci}.



Fig. 7. Temperature dynamics at 0.25 m depth at Happy Valley siteduring the summer of 2001 year. The graph shows that uncertaintyin temperature measurements is±0.02◦C. Within this uncertainty,the shadowed region represents a temperature range where the soilstarts to freeze. Therefore, the temperature,T∗, of freezing pointdepression lies within the shadowed regions, i.e. in [−0.04◦C/0◦C].

We continue this iterative processes until the difference be-tween the previous and current values of parameters inC isbelow a critical tolerance.

The selected periods1k do not have to coincide with tra-ditional subdivision of a year. The choice of1k is naturallydictated by seasons in the hydrological year, which starts atthe end of summer and consists several seasons. If the periodof observations is one year, typical intervals1k are “winter”,“summer and fall”, “fall” and “extended summer and fall”,see Table 2. We note that the intervals1k can overlap eachother, and quantitiests and te determining lower and upperlimits of integration in (26) are equal to the beginning andend of the time interval1k. For different geographical re-gions, the timing for the “winter”, “summer and fall” and“fall” can be different. Typical timing of periods{1k} forthe North Slope of Alaska is shown in Table 2, and are nowdiscussed.

5.2 Subproblems

11: The “winter” period corresponds to the time when therate of change of the unfrozen liquid water contentθl is neg-ligibly small; the heat Eq. (2) models the transient heat con-duction with thermal propertiesλ=λf , C=Cf , and dθl

dT≃0.

During the “winter”, temperature dynamics depend only onthe thermal diffusivityCf /λf of the frozen soil, and hencethe simultaneous determination of both parametersCf andλf is an ill-conditioned problem. Assuming that the heat

capacity{C(i)f } is known (depending on the soil texture andmoisture content we can approximate it using published

data), we evaluate the thermal conductivity{λ(i)f } and use

these values during minimization at other intervals.12: During the “summer and fall” time interval, active

phase change of soil moisture occurs. Hence, at this time,see Table 2, a contribution of the heat capacityC into theapparent heat capacityCapp is negligibly small comparingto the contribution of the latent heat termLdθl/dT . There-fore, the rate of freezing/thawing primarily depends on thesoil porosityη and the thermal conductivityλ (Tikhonov andSamarskii, 1963). Thus we approximate{C

(i)t } using pub-

lished data by analyzing the soil texture and moisture con-tent. Note that temperature-dependent latent heat effects dueto the existence of unfrozen waterθl at this period have a sec-ond order of magnitude effect (see discussion below). There-fore, if no prior information about the coefficientsb, T∗ pa-rameterizing the unfrozen water content is available then theycan be prescribed by taking into account the soil texture andanalyzing measured temperature dynamics at the beginningof freeze-up (see Fig. 7). We seek better estimates ofb, T∗ atthe next steps, namely during the “fall” period.

Since, during the “summer and fall” interval, the tempera-ture dynamics primarily depends on the porosityη and ther-mal conductivityλ, we have to find only{λ(j)t , η(j)}, since{λ(j)

f } are already found at the previous step, i.e. the “winter”interval. Taking into account the relationship (7) betweenthe thermal conductivities for completely frozen and thawedsoil, we approximate

λ(j)t = λ

(j)

f

[λl

λi

]η(j)

, j = 2, . . . , n. (27)

We remind that the water contentθl in the upper soil horizonchanges during the year due to moisture evaporation and pre-cipitation and is not always equal toη(1). Hence formula (27)does not hold forj=1. Hence, during the “summer and fall”period our goal is to estimate{η(i)}ni=1 andλ(1)t , and then de-

termine thermal conductivityλ(j)t for the rest of soil layersj=2, . . ., n using (27).13: Recall that while evaluating the thermal properties

{λ(i)t , λ

(i)f } and the soil porosity{η(i)}, we assumed that the

coefficients{b(i), T (i)∗ } are known. However they also haveto be determined. We remind that the coefficients{b(i), T

(i)∗ }

cannot be computed prior to calculation of{λ(i)t , λ

(i)f } and

{η(i)}, since{b(i), T (i)∗ } are related to the second order effectsin temperature dynamics during “summer and fall” and “win-ter” intervals. Once an initial approximation to{λ(i)t , λ

(i)f }

and {η(i)} is established, we consider the “fall” period (seeTable 2) during which the temperature dynamics strongly de-pends on{b(i), T (i)∗ } and allows us to capture second order ef-fects in temperature dynamics (Osterkamp and Romanovsky,1997).14: In the previous three periods, we obtained approxi-

mations to all variables{λ(i)t , λ(i)f , C(i)t , C(i)f , η(i), b(i), T (i)∗ }.



Table 2. Typical choice of parameters in the controlC for “cold” permafrost regions.

Periods Cj Typical1k Characteristic Step

“Winter” {λ(i)f

} December–April Completely frozen ground,T<−5◦C 1

“Summer and Fall” {η(i)},λ(1)t May–November Developing/-ed active layer and its freezing 2

“Fall” {b(i), T (i)∗ } September–December Active layer freezing,T>−5◦C 3

“Extended Summer and Fall” {η(i), T (i)∗ } May–January Developing/-ed active layer and its freezing 4

However, we can improve the approximation by consider-ing the “extended summer and fall” period, see Table 2.This period is associated with a time interval when the soilfirst thaws and then later becomes completely frozen. Sincepreviously, we minimized the cost function depending sep-arately on the porosity{η(i)} (“summer and fall”) and on{T

(i)∗ } (“fall”), we minimize the cost function depending si-

multaneously on{η(i)} and{T(i)∗ } during “extended summer

and fall”, while other parameters are fixed.We list in Table 2 all steps and time periods1k which

are necessary to find the initial approximation. One of thesequences of minimization steps is

“winter” → “summer and fall”→“fall” → “extended summer and fall”

From our experience with this algorithm, we conclude thatin some circumstances it is necessary to repeat minimizationover some time periods several times, e.g.

“winter” → “summer and fall”→“fall” → “extended summer and fall”→“fall” → “extended summer and fall”

until the consecutive iterations modify the thermal propertiesinsignificantly.

6 Application. Happy Valley site

6.1 Short site description

The temperature measurements were taken in the tussocktundra site located at the Happy Valley (69◦8′ N ,148◦50′ W)in the northern foothills of the Brooks Range in Alaska from22 July 2001 until 22 February 2005. We used data from 22July 2001 until 15 May 2002 to estimate soil properties, andfrom 15 May 2002 until 22 February 2005 to validate theestimated properties. The site was instrumented by eleventhermistors arranged vertically at depths of 0.02, 0.10, 0.17,0.25, 0.32, 0.40, 0.48, 0.55, 0.70, 0.86 and 1.06 m. The tem-perature sensors were embedded into a plastic pipe (the MRCprobe), that was inserted into a small diameter hole drilledinto the ground. The empty space between the MRC and theground was filled with a slurry of similar material to diminishan impact of the probe to the thermal regime of soil. Our frostheave measurements show that the vertical displacement ofthe ground versus the MRC probe is negligibly small at this

particular installation site. Prior to the installation, all sen-sors were referenced to 0◦C in an ice slush bath and havethe precision of 0.04◦C. An automatic reading of tempera-ture were taken every five minutes, then averaged hourly andstored in a data logger memory.

During the installation, soil horizons were described andtheir thicknesses were measured. The soil has three distincthorizons: organic cover, organically enriched mineral soil,and mineral soil. The boundaries between the horizons lie at0.10 and 0.20 m depth.

In the all following numerical simulations we consider aslab of ground representing the Happy Valley soil between0.02 and 1.06 m depth. For the computational purposes, theupper and lower boundary conditions are given by the ob-served temperatures at depth of 0.02 and 1.06 m. Also in allcomputations, the temperatures are compared with the set ofmeasured temperatures at the depths{xi}={0.10, 0.17, 0.25,0.32, 0.40, 0.48, 0.55, 0.70, 0.86} m.

6.2 Selection of an initial approximation

The “winter” period is associated to the ground temperaturebelow−5◦C, occurring on 15 January 2002 through 15 May2002 at the Happy Valley site. The heat capacityCf for eachlayer is evaluated based on the soil type, texture and is takenfrom (Hinzman et al., 1991; Romanovsky and Osterkamp,1995; Osterkamp and Romanovsky, 1996).

We estimateλf for each layer by looking for a minimum

of the cost functionJ in the 3-D space{λ(1)f , λ(2)f , λ

(3)f }.

The minimization problem in this space can be simplified bylooking for a minimum in the following series of 2-D prob-lems. For example, for several physically acceptable valuesof the thermal conductivityλ(1)f , we compute temperature dy-

namics for various values ofλ(2)f , λ(3)f and plot isolines of thecost functionJ . In the series of plots in Fig. 8, we noticethat a location of the minimum on the(λ(2)f , λ

(3)f ) plane shifts

asλ(1)f changes. The minimum of the cost function at eachcross section is almost the same, and the problem of selectingthe right combination of parameters arises. Here, knowledgeof the soil structure becomes relevant. It is known that thesoil type of third layer is silt highly enriched with ice, sofrom Table 1 1.6<λ(3)f <2.0. Therefore, we selectλ(1)f =0.55,

λ(2)f =1.0 andλ(3)f =1.8, and use them in all other consecutive



Fig. 8. The isolines of the cost functionJ on the plane(λ(2)f, λ(3)f) for different values of the thermal conductivityλ(1)

fkeeping constant

at each plot. The values ofλ(1)f

from the left to the right are 0.35, 0.55 are 0.70, respectively. The star in the central plot marks a selectedcombination of the thermal conductivities.

steps (see Table 3, columns 6,7 and 8). More precise resultscould be obtained if a sensor measuring the thermal conduc-tivity was placed in at least one of the horizons, see discus-sion in Sect. 7.

The “summer and fall” period is selected to capture themaximal depth of active layer occurring between 28 Au-gust 2001 and 6 December 2001. We take values of theheat capacityCt from (Hinzman et al., 1991; Romanovskyand Osterkamp, 1995; Osterkamp and Romanovsky, 1996).Comparing measured temperatures to the ones computedfor λ(1)t , {η(i)}3

i=1 varying within a range of their natural

variability, we found thatλ(1)t ∈[0.09,0.15], η(1)∈[0.3,0.9],η(2)∈[0.3,0.9] andη(3)∈[0.15,0.45]. Once the variability ofthese parameters is found, we search for a minimum of thecost function in the 4-D space{λ(1)t , η

(1), η(2), η(3)}, whereeach parameter varies within the found boundaries. We notethat during minimization ofJ in this 4-D space, other vari-ables inC are fixed and their values are listed in 1st “Summerand Fall” row in Table 3. For example, values of the thermalconductivityλ(1)f =0.55,λ(2)f =1.0 andλ(3)f =1.8 are obtainedat the previous step after minimization over the “winter” in-terval. Also, an approximation to the coefficientsb(i)=0.7,T(i)∗ =−0.03, i=1,2,3 in (3) is obtained by analyzing soil

texture and type, and dynamics of the measured temperaturesnear 0◦C, see Fig. 7. We emphasis that the approximation tothe parametersb andT∗ is tentative and is going to be im-proved at the consequent steps.

We note that it is not necessary to find a minimum in thefour dimensional space accurately but rather only to estimateits location as significant uncertainties in other parametersstill exist. Therefore, we look for the minimum by evaluatingthe cost function on(λ(1)t , η

(1)), (η(1), η(2)) and(η(2), η(3))planes as follows.

First, we setη(1)=0.6, η(2)=0.6, η(3)=0.3 andλ(1)t =0.12,which correspond to the middle of their variability ranges.Then, we evaluate the cost functionJ on the (λ(1)t , η

(1))

plane, by varyingλ(1)t , η(1) in the control, while all other

variables inC are fixed. In the left plot in Fig. 9, we plotisolines ofJ on all three planes(λ(1)t , η

(1)), (η(1), η(2)), and(η(2), η(3)).

At the (λ(1)t , η(1)) plane, the cost function attains its min-

imal value on a boundary of this plane, see Fig. 9, upperleft, and is minimal in the center of the planes(η(1), η(2)),(η(2), η(3)). The last two planes allows us to find thatη(1)=0.6, η(2)=0.55 andη(3)=0.27, whereas contours at thefirst plane show that the value ofλ(1)t lies between 0.11 and0.13, see Fig. 9, left column. We suppose thatλ

(1)t is 0.12

and proceed further. After updating the control with the com-puted values, we evaluate the cost function on the same setof planes one more time; parameters in the control beforeminimization are shown in Table 3 the “Summer and Fall”2nd row. After computing the cost function, we draw its iso-lines and show them in Fig. 9, right. Note that at this stepthe cost function attains its minima located in the center ofthe computational grid. We update the control withη(1)=0.6,η(2)=0.55, η(3)=0.27, λ(1)t =0.12. Note that the location ofthe minimum did not change significantly. Our experienceshows that changes of soil properties by 5%–10% or less areinsignificant, since the corresponding difference in soil tem-peratures is comparable with uncertainties of measurements.Therefore, we do not have to do additional iterations on thesame set of planes, and we proceed to the next step and re-duce uncertainties in coefficientsT∗ andb.

For the sake of brevity, we omit details in consequent stepsassociated with “fall” and “extended summer and fall” inter-vals, since the search of parameters is completed similar to asdescribed in the “summer and fall” step. We emphasize thatwe are just interested in calculation of an initial approxima-tion to the control which could serve as a starting point in theglobal minimization of the cost function. By no means, dowe try to substitute the global minimization by this heuristicprocedure. However, a good starting point can save compu-tational time and improve accuracy of a final result.



Table 3. Values of parameters in the control at the beginning of each minimization step. The listed steps are typical to recover the initialapproximation to the soil properties. The parameters which values are in the parenthesis with the same subindex define minimization plane.

For example, in the third rowλ(1)t andη(1) are in the parenthesis and have the same subindex equal to 1. Therefore, this pair define a

minimization plane(λ(1)t , η(1)). On this plane we minimize the cost function depending onλ(1)t andη(1), while value of other parameters

are fixed and given in other sections of the current row.

Iterations η(1) η(2) η(3) λ(1)t λ

(1)f

λ(2)f

λ(3)f

b(1) b(2) b(3) T(1)∗ T

(2)∗ T

(3)∗

Winter 0.40 0.70 0.25 0.10 – – – 0.7 0.7 0.7 −0.03 −0.03 −0.03Summer and Fall, 1st (0.60)1,2 (0.60)2,3 (0.30)3 (0.12)1 0.55 1.00 1.80 0.7 0.7 0.7 −0.03 −0.03 −0.03Summer and Fall, 2nd (0.60)1 (0.55)1,2 (0.27)2,3 (0.12)3 0.55 1.00 1.80 0.7 0.7 0.7 −0.03 −0.03 −0.03Fall, 1st 0.60 0.55 0.27 0.12 0.55 1.00 1.80 (0.7)1 (0.7)2 (0.7)3 (−0.03)1 (−0.03)2 (−0.03)3Fall, 2nd 0.60 0.55 0.27 0.12 0.55 1.00 1.80 (0.7)1 (0.6)2 (0.75)3 (−0.03)1 (−0.03)2 (−0.03)3Ext. Summer and Fall 1st (0.60)1 (0.55)2 (0.27)3 0.12 0.55 1.00 1.80 0.65 0.6 0.75 (−0.02)1 (−0.03)2 (−0.03)3Ext. Summer and Fall 2nd (0.70)1 (0.55)2 (0.27)3 0.12 0.55 1.00 1.80 0.65 0.6 0.75 (−0.025)1 (−0.03)2 (−0.03)3

Final Result 0.70 0.55 0.27 0.12 0.55 1.00 1.80 0.65 0.6 0.75 −0.025 −0.025 −0.03

6.3 Global minimization and sensitivity analysis

While evaluating an initial approximation, we sought minimaof the cost functionsJ (C) measuring discrepancy over peri-ods{1k}. In this subsection, we perform global minimiza-tion of the cost function with respect to all parameters inC

over the entire period of measurements 22 July 2001 until 15May 2002 used for calibration. Also, we analyze sensitivityof an initial approximation derived from minimizing the costfunction globally with respect to all parameters.

In global minimization problems, a starting point fromwhich iterations begin is given by the initial approximationevaluated in the previous subsection, see Table 3, the lastrow. As a result of global minimization problem, we obtainthe parameters (thermal properties, porosity and coefficientsspecifying the unfrozen water content for each soil horizon)which can depend on valuests andte determining the periodover which discrepancy between observed and modeled tem-peratures is measured. In global minimization problems, theconstantte is associated with an end of “winter” interval dur-ing which the soil is completely frozen. But since, the soilis frozen for several months for a cold permafrost region, thecost function does not significantly depends onte if te varieswithin two week limits. However, the value ofts is asso-ciated with beginning of “summer and fall” interval duringwhich the ground is thawed. Since, the ground is thawingduring a relatively short period of time for cold permafrostregions, we consider several values ofts and minimize thecost function with respect to all parameters.

Results of minimization are listed in Table 4. It shows thatthe results of global minimization do not significantly dependon constantsts , if the interval[ts, te] represents thawed andfrozen states of the soil. Using averaged values of the ther-mal properties, we compute the temperature dynamics for theentire period of observations. Comparison of the calculatedand measured temperatures at different depths and at time in-tervals used for calibration are shown in Figs. 10, 11 and 12.During the winter, the calculated temperature closely follows

the observed temperature within the uncertainty of thermistormeasurements. During the summer, the difference betweenthe measured and calculated temperatures is larger but doesnot exceed 0.3◦C for sensors in the mineral soil. This largerdiscrepancy between the measured and computed tempera-tures can be partially explained by over-simplifying physicsand neglecting water dynamics in the upper organic horizons.

Finally, in order to show that the found initial approxima-tion (the last row in Table 3) lies close to the true values ofsoil properties, we use it to compute the soil temperature dy-namics through 22 February 2005. Note that the time intervalfrom 22 May 2002 until 22 February 2005 was not used tofind the soil properties. In Fig. 13, we plot the measuredand calculated temperature dynamics at 0.55 m depths. Theplots with solid symbols mark temperature dynamics com-puted for the found initial approximation, and for the bestguess values (in the middle of the variability range, shown inTable 1). We note that the guessed values are used to providebenchmark temperature dynamics against which we show ef-fectiveness of our approach. The benchmark temperature ismuch warmer during summer, and the freeze-up occurs sev-eral days later than in the measured temperature. The bench-mark temperature dynamics during winter closely followsthe measured temperature dynamics, since the middle of thevariability range, for which the benchmark temperature wascomputed, almost matches the found initial approximation.The difference between the measured temperature dynamicsand the one calculated for the found initial approximation istypically less than 0.25◦C.

7 Discussion and limitation of the proposed method

We concentrate on finding soil thermal properties by mini-mizing the multivariate cost functionJ . There are many wellknown methods that find a minimum ofJ , including stochas-tic, heuristic and gradient type algorithms (Goldberg, 1989;Fletcher, 2000; Robert and Casella, 2004). Since we focuson a gradient type algorithm, a special care is necessary to



Fig. 9. Selection of the thermal conductivityλ(1)t and the soil porosityη(1), η(2), η(3) by minimizing the cost function associated with the“summer and fall” interval. The left and right column are associated with the first and the second iterations, respectively. The stars markselected values of parameters after completing the iteration. Note that at the second iteration stars and locations of all minima are coincide.

select the initial approximation to the soil properties. For ex-ample, if the gradient type algorithm is started outside thebasin of attraction of the proper minimum, then due to exis-tence of multiple local minima it can converge to physicallynon-realistic combination of parameters inC. Stochastic andheuristic algorithms can possibly avoid the problem of select-

ing the initial approximation, since they are not get trappedin a neighborhood of the local minimum. However, a qual-ity control of soil properties recovered by either stochasticor heuristic algorithm arises. For example, it is feasible thatseveral local minima can have the same value and correspondto substantially different combinations of parameters in the



Table 4. Global minimization with respect to all parameters in the control. Each realization is specified by the time interval[ts , te] overwhich the discrepancy between the data and computed temperature dynamics is evaluated. In all case, the constantte is 15 May 2002.

ts η(1) η(2) η(3) λ(1)t λ

(1)f

λ(2)f

λ(3)f

b(1) b(2) b(3) T(1)∗ T

(2)∗ T

(3)∗

August,18 0.703 0.560 0.272 0.120 0.562 0.983 1.797 0.655 0.596 0.750 −0.0251 −0.0253 −0.0301August,22 0.721 0.557 0.272 0.122 0.559 0.973 1.809 0.673 0.558 0.757 −0.0256 −0.0249 −0.0295August,26 0.718 0.546 0.272 0.122 0.559 0.962 1.801 0.657 0.597 0.755 −0.0250 −0.0251 −0.0303August,30 0.712 0.549 0.272 0.121 0.556 0.967 1.801 0.655 0.601 0.755 −0.0251 −0.0251 −0.0302September,3 0.712 0.544 0.274 0.123 0.559 0.980 1.816 0.665 0.551 0.750 −0.0255 −0.0255 −0.0298September,7 0.718 0.534 0.274 0.123 0.560 0.966 1.789 0.660 0.603 0.747 −0.0250 −0.0252 −0.0297

Fig. 10. Measured (hollow) and calculated (solid) temperature at0.10, 0.17 and 0.25 m depth. The time interval is associated withthe “summer and fall” period.

control vectorC. Thus, straight forward application of thesealgorithms can result in soil properties that are different fromthe physically realistic soil properties several fold.

In this article, we find an initial approximation to the ther-mal properties. This initial approximation can be later usedin gradient type algorithm both as a starting point and a regu-larization. We admit that we find one of the possible realiza-tions for the initial approximations. However, in the processof its computation, we obtain limiting boundaries on param-eters inC which can constrain multivariate minimization, in-dependent on the type of algorithm, i.e. stochastic, heuristic,or the gradient type.

We describe a technique to find an initial approximationto the thermal properties of soil horizons. This technique ap-proximates the thermal conductivity, porosity, unfrozen wa-ter content curve in horizons where no direct temperaturemeasurements are available. One of the limitations is thatit requires values of heat capacities, since at certain time pe-riods it is possible to estimate thermal diffusivity only but notthermal conductivity and heat capacity separately.

Fig. 11. Measured (hollow) and calculated (solid) temperature at0.32, 0.48, and 0.70 m depth. The time interval is associated withthe “winter” period.

Since, due to a short distance between points at which theupper and lower boundary conditions are specified, there isuncertainty in evaluation of the thermal conductivity for thefrozen ground. This uncertainty is related to a progressive lagin the phase of the temperature wave, defined by the SecondFourier Law (Carslaw and Jaeger, 1959). The longer the lag,the better the thermal conductivity can be estimated. For theHappy Valley site, the temperature lag at 1 m depth is about10 days. Our experience suggests that the 20–30 day time lagis adequate to estimate thermal conductivity robustly. In thecase of shorter time lags, we advocate placing of a thermalconductivity sensor in the mineral soil horizon. The thermalconductivity sensor consists of a heating element and a ther-mocouple embedded in a needle. More information regard-ing the sensor can be found in (Thermal Logic, 2001) andin references therein. One of the limitations on usage of thethermal conductivity sensor is that it generates correct valuesof the thermal conductivity for thawed or completely frozensoil in which active phase change processes do not occur.



Fig. 12. Measured (hollow) and calculated (solid) temperature at0.55, 0.70 and 0.86 m depth during the entire period of measure-ments used for calibration.

It should be noted that recovery of the thermal propertiesof the organic cover (e.g. moss layer) is given as an inte-grated approach in the following sense. Complex physicalprocesses occurring in the organic cover that include non-conductive heat transfer (Kane et al., 2001) are taken into ac-count by estimating some effective thermal properties whichare constants for the entire season. We acknowledge that theestimated thermal properties of the organic layer could bedifferent in nature, but we recover them in such a way that thetemperature in the active layer and permafrost should corre-spond to the measured one.

In the proposed model we used 1-D assumption regard-ing the heat diffusion in the active layer, which sometimes isnot applicable due to hummocky terrain in the Arctic tundra.Another assumption used in the model is that frost heave andthaw settlement is negligibly small and there is no ice lensformation in the ground during freezing. Therefore, the pro-posed method could be only applied where these assumptionare satisfied.

The proposed method allows computation of a volumetriccontentη of water which changes its phase during freezing orthawing. Water content of liquid water that is tightly boundto soil particles and is not changing its phase can not be esti-mated, see Fig. 1.

8 Conclusions

We present a technique to calculate an approximation to thesoil thermal properties, porosity, and parametrization of theunfrozen water content in order to use it in gradient type it-erative minimization methods both as a starting point and asa regularization. To compute the approximation, we min-imize the multivariate cost function describing discrepancy

Fig. 13. Measured (hollow) and calculated (solid) temperature at0.55 m depth during the entire period of measurements. The val-idation represents temperature dynamics computed for the foundapproximation to the soil thermal properties. The benchmark repre-sents temperatures computed for the best guess soil properties, i.e.in the middle of ranges listed in Table 1.

between the measured and calculated temperatures over acertain time interval. We find the minimum by adopting acoordinate-wise iterative search technique to the specifics ofour inverse problem. At each iteration, we select a particularset of soil properties and associate with them a certain timeinterval over which we minimize the cost function. Afteremploying the proposed sequence of iterations, it is possibleto find the approximation to all thermal properties and soilporosity.

Although there are several limitations to the proposed ap-proach, we applied it to recover soil properties for HappyValley site near Dalton highway in Alaska. The differencebetween the simulated and measured temperature dynamicsover the periods of calibration is typically less than 0.3◦C.The difference between the simulated and measured temper-atures over the consecutive time interval not used in calibra-tion is less than 0.5◦C which shows a good agreement withmeasurements, and validates that the found initial approxi-mation lies close to the true values of soil properties.

In order to compute the cost function, it is necessary tocalculate the soil temperature dynamics. Therefore, we de-veloped a new finite element discretization of the Stefan-typeproblem on fixed coarse grids using enthalpy formulation.One of the advantages of the new method is that it allowscomputation of the temperature dynamics for the classicalStefan problem without any smoothing of the enthalpy. Also,new approach shows equal or better performance compar-ing to other finite element models of the ground thawing andfreezing processes.



Acknowledgements. We would like to thank J. Stroh, S. Mar-chenko, A. Kholodov, and P. Layer for all their advice, critiqueand reassurances along the way. We are thankful to reviewersand the editor for valuable suggestions making the manuscripteasier to read and understand. This research was funded byARCSS Program and by the Polar Earth Science Program, Officeof Polar Programs, National Science Foundation (OPP-0120736,ARC-0632400, ARC-0520578, ARC-0612533, IARC-NSF CA:Project 3.1 Permafrost Research), by NASA Water and EnergyCycle grant, and by the State of Alaska.

Edited by: S. Gruber

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